Variability of Soil-Structure System Frequencies during ... · Maria I. Todorovska,a) Tzong-Ying...

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Variability of Soil-Structure System Frequencies during Strong Earthquake Shaking for a Group of Buildings in Los Angeles Estimated from Strong Motion Records

Maria I. Todorovska,a) Tzong-Ying Hao,a) and Mihailo D. Trifunaca)

Most seismic building codes estimate the design forces in structures based on

the seismic coefficient C(T), where T is the “fundamental vibration period of the

building.” For structures on flexible soil, T is the first period of the soil-structure

system, which depends on the structure itself, but also on the foundation system,

surrounding soil, and contact conditions between the foundation and the soil.

This paper presents results of instantaneous system frequency as function of the

level of response for seven buildings in the Los Angeles area that have recorded

several earthquakes—1994 Northridge earthquake (MS= 6.7) and aftershocks, and

1971 San Fernando earthquake (MS = 6.6). In general, the observed trend is

decrease of system frequency during 1994 Northridge and 1971 San Fernando

earthquakes, which caused the largest levels of response, and “recovery” during

the aftershocks. For one of the buildings, a significant change (30% reduction)

that occurred during the San Fernando earthquake appears to have been

permanent. For most buildings, the frequency changed up to 20%, and for two

buildings, the change was about 30%. Understanding and estimation of the range

of these variations during strong earthquake shaking is needed for further

refinement of the existing and development of new design code procedures.

INTRODUCTION

The Earthquake Resistant Design Codes have evolved based on principles and

procedures derived from the Response Spectrum Method (Biot, 1942). In most codes, the

design shear forces are quantified using the seismic coefficient C(T), where T is the

“fundamental vibration period of the building,” and various scaling factors that depend on the

seismic zone, type of structure, soil site conditions, importance of the structure etc. As T

a) Civil Engineering Department, University of Southern California, Los Angeles, CA 90089-2531

Proceedings Third UJNR Workshop on Soil-Structure Interaction, March 29-30, 2004, Menlo Park, California, USA.

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cannot be measured before the structure is completed, most codes provide simplified

empirical formulae to estimate it, based on past experience and recorded response of existing

buildings, which is extremely limited (both in quantity and in quality). The problem of

estimation of T has been considered by many investigators, based on theory (Biot, 1942),

small amplitude ambient and forced vibration tests of full-scale structures (Carder, 1936), and

recorded earthquake response in structures (Li and Mau, 1979). Unfortunately, the number

of well-documented instrumented buildings that have recorded at least one earthquake is

typically less than 100. When the recorded data is grouped by structural systems (moment

resistant frame, shear wall etc.) and building materials (reinforced concrete, steal, etc.), the

number of records per group becomes too small to control the accuracy of regression

analyses, or to separate “good” from “bad” empirical models (Goel and Chopra, 1997;

Stewart et al., 1999). This problem is further complicated by the nonlinearity of the

foundation soil even for very small strains (Hudson, 1970; Luco et al., 1987). During strong

earthquake shaking, the apparent period of the soil-foundation-structure system, T , can

lengthen significantly (Udwadia and Trifunac, 1974), and it may or may not return to its pre

earthquake value. This period lengthening can reach and exceed a factor of two, which adds

to the scatter in the empirical regression analyses, and to the ambiguity in choosing a

representative T for evaluation of C(T) (Trifunac, 1999; 2000).

For further improvements and developments of the building codes, it is essential to

understand the amplitude dependent period lengthening (as function of the level of response

of the structure and strain in the soil), and estimate its range, which can be best accomplished

by analysis of building periods from multiple earthquake recordings in buildings—of both

small and large levels of shaking. The first step towards this goal is to augment the database

of multiple earthquake records in buildings, which is very limited, because most buildings

records have been recorded on film, and, mostly those with larger amplitudes have been

digitized and released. Data of small amplitude response will be generated fast from

instrumented buildings with a digital recording system, but it may take many years for these

systems to record larger amplitude response. Hence, the use of small amplitude data from

newly instrumented buildings is quite limited. Data of smaller amplitude response has been

recorded digitally in some buildings complementing larger amplitude response data recorded

previously by an analog recording system. In these cases, smaller amplitude analog

recordings of past earthquakes are also very valuable—for understanding of the variations of

building periods with time, which may be temporary or permanent. In the Los Angeles

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metropolitan area, there have been many small earthquakes and aftershocks of larger

earthquakes that have been recorded in buildings and archived but not digitized and released.

An effort was initiated at the University of Southern California to augment the database

of building periods estimated from multiple earthquake recordings, with the immediate

objective to trace their variations with time and as a function of the level of response and

understand their nature, and with the ultimate objective—to improve the code formulae for

estimation of building periods. This paper presents a summary of results of a one year effort,

which consisted of digitization and processing of building records from the archives of the

U.S. Geological Survey (USGS), gathering of already processed data for the same buildings,

and analysis of the building period. Results are shown of the variation of the building

apparent frequency (i.e. building-foundation-soil system frequency) as a function of the level

of response for seven buildings in the Los Angeles area, which recorded the Northridge

earthquake and some of its aftershocks. The records in these seven buildings that have been

digitized and processed for this project are those of aftershocks of the 1994 Northridge

earthquake as well as those of the main event that have not been digitized previously or have

been redigitized for this project. These buildings have been instrumented either by USGS or

by the building owner. Detailed results and a catalog of the processed data can be found in

Todorovska et al. (2004) and the processed data is available from the USC Strong Motion

Research Group web site at www.usc.edu/dept/civil_eng/earthquake_eng/.

STRONG MOTION DATA

Figure 1 shows a map of the Los Angeles metropolitan area and locations of buildings

that were instrumented at the time of the 1994 Northridge earthquake, either by the USGS

and partner organizations, or by the building owners (as required by the Los Angeles and

state building codes), for which the data is archived by USGS. The latter are often referred to

as “code” buildings. All of these buildings will be referred to as “USGS instrumented

buildings” and identified by their station number.

The sensors in these buildings have been either three-component SMA-1 or multi-

channel CR-1 accelerographs, both recording on film. Many of the “code” buildings (about

30 buildings total) have only one instrument, at the roof, due to a change in the original

ordinance for Los Angeles, such that only one instrument at the roof was required, which

lead to removal or neglect of the instruments at the ground floor and intermediate levels.

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This unfortunate fact limits considerably the use of these records, especially for analyses of

soil-structure interaction. The roof records can be used to estimate the apparent building

period, by approximating the relative roof motion (with respect to the base) by the absolute

roof motion.

After the Northridge earthquake, the analog strong motion instrumentation is being

gradually replaced by digital, and additional buildings are being instrumented. For some of

these buildings, data of smaller local earthquakes and distant larger earthquakes has been

recorded and released. The recorded level of response for these events, however, is much

smaller than that for the Northridge earthquake. Figure 1 also shows the epicenters of

earthquakes that have been recorded in these buildings. The Northridge main event was

followed by a large number of aftershocks (9 of these had M > 5, and 55 had M > 4). Many

of these larger magnitude aftershocks, as well as smaller magnitude but closer aftershocks,

were recorded in the instrumented buildings. The aftershock of March 20, 1994 (M = 5.2;

“aftershock 392”) was the one recorded by the largest number of (ground motion) stations

(Todorovska et al., 1999). The Northridge sequence was recorded on multiple films,

archived separately. The largest number of recorded aftershocks known to the authors of this

paper is 86—at station USGS #5455, and about 60 at several other stations. Unfortunately, it

turned out that the number of aftershock records useable for estimation of the building

apparent frequency was small—up to 11.

The usability of an aftershock record for estimation of the first system frequency

depends not only on its amplitudes, but also on the shape of its Fourier spectrum, and on the

building itself, i.e. on how high is its first system frequency. In principle, a building record is

useable if the first system frequency is within the “useable” frequency band of the signal,

which is the band where the Fourier spectrum of the signal is above that of the recording and

digitization noise, and below the Nyquist frequency. While the high-frequency limit of this

band is controlled by the Nyquist frequency, the low-frequency limit is controlled by the

noise spectrum, which increases with decreasing frequency and is due to a “wavy” baseline.

These limits are determined automatically by standard software for accelerogram data

processing (Lee and Trifunac, 1990), which then band-pass filters the digitized accelerogram.

In this study, if the first system frequency was too close to, or was suspected to be below the

low-frequency cut-off of the data processing software, the record was discarded. A record

was also discarded if the system frequency could not be estimated reliably for some other

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reason. This selection process eliminated many aftershock records. In summary, small

aftershock records are more likely to be useable for the shorter buildings, than for the taller

ones. Also, for the taller buildings, the records from large but distant earthquakes (like 1992

Landers) are more likely to be useable than those from small nearby events with similar peak

acceleration, because the former have more energy in the shorter frequency part of the

spectrum and will excite more the first mode, leading to larger signal to noise ratio at low

frequencies. Therefore, it is important to digitize and add to the analysis “good” records of

the 1992 Landers and 1999 Hector Mine earthquakes.

Los Angeles

Santa Monica

San Pedro

San Fernando Valley

San Gabriel Mountains

Pacific Ocean

5108

54515455

466

54575260

793

52845277

5450

5259 742

872572

5106

5465

5462

5258229

804

5239

482

5293892

5459437

52335292982

50825456

6635263

5453

5264

530

5460

Malibu, 1989 Montebello, 1989

Long Beach1933

S. California1989

Sierra Madre, 1991

Pasadena, 1988

Northridgeand Aftershocks1994 Whittier Narrows

and Aftershocks1987

San Fernando1972

West Hollywood2001

34.25

34.00

118.75 118.25118.50 118.00 117.75

34.50

33.75

33.50

USGS instrumeneted building sitesanalysis completed and reportedanalysis to be completed after additional data is processed

other USGS instrumeneted buildings

0 10 20

km

Figure 1. Locations of instrumented buildings in the Los Angeles metropolitan area at the time of the 1994 Northridge earthquake, for which the data is archived by USGS The building sites are identified by their USGS station number.

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This paper shows results for 7 buildings for which there were three or more adequate

records (of the Northridge sequence or of the 1971 San Fernando earthquake) to estimate the

apparent building frequency, and for which there are no other “good” records to add to the

analysis, so that the analysis is “complete.” These stations are marked by solid dark dots in

Fig. 1. For the 15 stations marked by solid light dots, there are some adequate records of the

Northridge sequence, as well as other “good” records that have not yet been digitized (e.g. of

the Landers and/or of the Whittier-Narrows earthquake). For the other buildings, marked in

Fig. 1 by solid rectangles, at this time, only one or maybe two “adequate” records for such

analysis are known to exist, and are not included in this analysis.

Table 1 shows a list of earthquakes recorded in “USGS” instrumented buildings. For the

Northridge sequence, only the aftershocks are shown for which there is an adequate record

that has been used in the analysis presented in this paper. For most of the buildings, the

contributing aftershocks have not been identified, but are assigned negative aftershock

number, the absolute value of which increases chronologically, and is related to the order of

the record on the film (e.g., aftershock –3 means that this was the third aftershock record on

the film following the main event, and aftershock –105 means that this was the fifth record

on the second role of film, which did not contain the main event). This table also lists the

2001 West Hollywood earthquake (M = 4.2), which occurred close to many of the

instrumented buildings (see Fig. 1), and which should have been recorded by these buildings.

Table 2 shows a list of buildings included in the analysis in this paper. The processed data

(uncorrected acceleration, corrected acceleration, velocity and displacement, and Fourier and

response spectra) is available free of charge from the USC Strong Motion Research Group

web site at www.usc.edu/dept/civil_eng/earthquake_eng/.

METHODOLOGY

The instantaneous frequency was estimated by two methods: (a) zero-crossing analysis, and

(b) from the ridge of the Gabor transform, both applied to the relative roof displacement

when there was a record at the base, or to the absolute displacement when only the roof

response was recorded, and considered as an approximation of the relative displacement in

the neighborhood of the first system frequency. Both methods were applied to the filtered

displacement, such that it contained only motion in the neighborhood of the first system

frequency, and resembled a chirp signal. The zero-crossing analysis consists of measuring

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Table 1. Earthquakes recorded by USGS instrumented buildings (1971 to 2001).

Event Date Time ML Latitude Longitude Depth (km)

San Fernando 02/09/1971 06:00 6.6 34 24 42N 118 24 00W -- Whittier-Narrows 10/01/1987 14:42 5.9 34 03 10N 118 04 34W 14.5 Whittier-Narrows, 12th Aft. 10/04/1987 10:59 5.3 34 04 01N 118 06 19W 13.0 Whittier-Narrows, 13th Aft. 02/03/1988 15:25 4.7 34 05 13N 118 02 52W 16.7 Pasadena 12/03/1988 11:38 4.9 34 08 56N 118 08 05W 13.3 Malibu 01/19/1989 06:53 5.0 33 55 07N 118 37 38W 11.8 Montebello 06/12/1989 16:57 4.4 34 01 39N 118 10 47W 15.6 Upland 02/28/1990 23:43 5.2 34 08 17N 117 42 10W 5.3 Sierra Madre 06/28/1991 14:43 5.8 34 15 45N 117 59 52W 12.0 Landers 06/28/1992 11:57 7.5 34 12 06N 116 26 06W 5.0 Big Bear 06/28/1992 15:05 6.5 34 12 06N 116 49 36W 5.0 Northridge 01/17/1994 12:30 6.7 34 12 48N 118 32 13W 18.4 Northridge, Aft. #1 01/17/1994 12:31 5.9 34 16 45N 118 28 25W 0.0 Northridge, Aft. #7 01/17/1994 12:39 4.9 34 15 39N 118 32 01W 14.8 Northridge, Aft. #9 01/17/1994 12:40 5.2 34 20 29N 118 36 05W 0.0 Northridge, Aft. #100 01/17/1994 17:56 4.6 34 13 39N 118 34 20W 19.2 Northridge, Aft. #129 01/17/1994 20:46 4.9 34 18 04N 118 33 55W 9.5 Northridge, Aft. #142 01/17/1994 23:33 5.6 34 19 34N 118 41 54W 9.8 Northridge, Aft. #151 01/18/1994 00:43 5.2 34 22 35N 118 41 53W 11.3 Northridge, Aft. #253 01/19/1994 21:09 5.1 34 22 43N 118 42 42W 14.4 Northridge, Aft. #254 01/19/1994 21:11 5.1 34 22 40N 118 37 10W 11.4 Northridge, Aft. #336 01/29/1994 11:20 5.1 34 18 21N 118 34 43W 1.1 Northridge, Aft. #392 03/20/1994 21:20 5.2 34 13 52N 118 28 30W 13.1 Hector Mine 10/16/1999 09:46 7.1 34 36 00N 116 16 12W 3.0 West Hollywood 09/09/2001 23:59 4.2 34 04 30N 118 22 44W 3.7

Table 2. USGS instrumented buildings included in the analysis.

USGS: 0466, SMA-1 185 Los Angeles, 15250 Ventura Blvd., Roof (13th floor) 34.157°N, 117.476°W

USGS: 5108 SMA 1276 and 1277 Canoga Park, Santa Susana, ETEC Bldg 462 (6th and 1st floors) 34.230°N, 118.712°W

USGS: 5450, SMA-1 6146 Burbank, 3601 West Olive Ave., Roof (9th floor) 34.152°N, 118.337°W

USGS: 5451, SMA-1 4048 Los Angeles, 6301 Owensmouth Ave., Roof (12th level) 34.185°N, 118.584°W

USGS: 5453, SMA-1 7073 Los Angeles, 5805 Sepulveda Blvd., Roof (9th floor) 34.175°N, 118.465°W

USGS: 5455, SMA-1 4270 Los Angeles, 16000 Ventura Blvd., Roof (13th floor) 34.156°N, 118.480°W

USGS: 5457, SMA 5491 LOS ANGELES, 8436 WEST 3rd ST., Roof (10th floor) 34.072°N, 118.375°W

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the time between consecutive zero crossings of the displacement, and assuming this time

interval to be a half of the system period (see Trifunac et al., 2001a, 2001b, 2001c). The

Gabor transform is a time-frequency distribution, which is up to a phase shift identical to a

moving window analysis with a Gaussian time window. The instantaneous frequency was

determined from the ridge of the transform, and the corresponding amplitude was estimated

from the skeleton of the transform, which is the value of the transform along the ridge (see

Todorovska, 2001). The wavelet transform with the complex Morlet wavelet was initially

considered, which is essentially a Gabor transform with a window that varies depending on

the frequency, so that it always contains same number of wavelengths. The results by both

methods were found to be very similar, and no advantage was seen in using a variable

window because the changes of the building frequency are relatively small, much smaller

than an order of magnitude. Using constant window was convenient in the estimation of the

resolution of the method. The Gabor transform was used with spread 1.5σ = . Figure 2

shows the Gabor wavelet and its Fourier transform.

-4 -3 -2 -1 0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

0.6 0.8 1.0 1.2 1.40

1

2

3

2σ t2σ f

Gabor wavelet, ω=2π, σ=1.5

g(b,ω)(t)

ξ/(2π) − Hzt − s

g(b,ω)(ξ)

Figure 2 A Gabor wavelet for time shift b = 0 and ω = 2π, in the time domain (left) and in the frequency domain (right).

The methodology for estimation of the instantaneous frequency of building-soil systems

is illustrated in Fig. 3, parts a and b, respectively for two records—component N11E of the

record of the 1971 San Fernando earthquake at station USGS 466 (Los Angeles, 15250

Ventura Blvd.), and component N00E of the record of the 1994 Northridge earthquake at

station USGS 5453 (Los Angeles, 5805 Sepulveda Blvd.). For each record, the plot on the

left hand side shows the Fourier spectrum of the relative roof displacement (solid line), or its

approximation by the absolute displacement when only a roof record was available, the

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Fourier spectrum of acceleration at the ground floor (dashed line)—if available, and a smooth

approximation of the relative (or absolute) roof displacement spectrum by the marginal

Gabor transform distribution (the smooth line). There are three plots in the right hand side,

as follows. The plot on the top shows the time history of the roof relative (or absolute)

displacement (solid line), and of the ground floor displacement (dashed line) if available, for

the “broad-band” data, which is the output of the standard data processing. The plot in the

middle shows the same time histories but for the “narrow-band” data, which is the broad-

band data filtered so that it contains only the frequencies in the neighborhood of the first

building-soil system frequency. The cut-off and role-off frequencies, in Hz, of the Ormsby

filters used are shown in the upper right corner of these plots. The plot in the bottom shows

the instantaneous frequency versus time estimated by the zero-crossing analysis (open

circles), and Gabor analysis (with 1.5σ = for all the records). The shaded rectangle in this

plot has width 2 tσ and height 2 ωσ and is a measure of the resolution of the Gabor analysis.

The Gabor transform at a point ( ),t f in the time frequency plane is the weighed average of

the components of the function (effectively) within such a rectangle centered at that point.

The method cannot resolve frequencies that are closer than ωσ , and estimates in time that are

closer than tσ . The resolution in frequency can be increased only if the resolution in time is

decreased (by increasing the time window of the Gabor wavelet, and consequently— tσ ), and

vice versa.

The results in Fig. 3 show that the estimates by the zero-crossing and Gabor analyses are

consistent. The estimates by the latter method are smoother, as the Gabor transform is a

smoothing operator. The zero-crossing analysis is not accurate when the oscillations of the

signal depart too much from a “pure” harmonic, and these estimates are not shown. Both

methods are most accurate when the amplitude of the signal is large and does not vary

significantly during one cycle of oscillation, least accurate—when the amplitude is small and

varies significantly during one cycle, and are arbitrary when the amplitude is practically zero.

A significant change (decrease) in the system frequency of about 30% during a single

earthquake is seen for both buildings.

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Figure 3. Estimation of the instantaneous frequency for a) component N00E of the record of the 1994 Northridge earthquake at station USGS 5453 (Los Angeles, 5805 Sepulveda Blvd.), and b) component N11E of the record of the 1971 San Fernando earthquake at station USGS 466 (Los Angeles, 15250 Ventura Blvd.).

a)

b)

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RESULTS

Figures 4 through 10 show, in a compact form, results for the variation of the building-

soil system frequency in time as function of the amplitude of response for the seven

buildings. Detailed results, like those in Fig. 3, for every earthquake and component of

motion, can be found in Todorovska et al. (2004). Each of these figures shows four plots, as

follows. Those on the top correspond to one of the two horizontal component of motion, and

those on the bottom—to the other horizontal component of motion, while those on the left

correspond to estimates by zero-crossing analysis, and those on the right—by Gabor analysis.

In each plot, the horizontal axis corresponds to the instantaneous frequency, the vertical axis

corresponds to the amplitude of response (of the filtered signal) expressed as a rocking angle

in radians, and each point corresponds to a particular instant in time. The rocking angle was

computed by dividing by the distance between the top and bottom instruments, estimated

using average floor height of 12.5 feet (1 foot=30.48 cm) the amplitude of the relative (roof

minus base) response, if motion at the base was recorded, or otherwise—the absolute

horizontal response of the roof or top floor. It is noted here that this rocking angle includes

the rigid body rocking, which could not be separated because of insufficient number of

instruments at the base, in addition to motion resulting from deflection of the structure.

In Figs 4 through 10, the points corresponding to consecutive instants of time are

connected by a line, each line corresponding to a particular earthquake. In the plots showing

results from the zero-crossing analysis, the first and last point shown for a particular

earthquake are marked respectively by an open and a closed circle. The backbone curve,

drawn by hand, indicates roughly the trend of the variation of the system frequency as

function of the amplitude of response. It can be seen that for the largest motions (during the

1971 San Fernando and 1994 Northridge earthquakes), the system frequency generally

decreased during the shaking. For all but one building, this change seems to have been

temporary, as the system frequency increased during the shaking by the aftershocks. For one

building, permanent change appears to have occurred during the 1971 San Fernando

earthquake (USGS 466). Detailed interpretation of the causes of these changes is beyond the

scope of this project. The maximum and minimum frequencies determined from the

backbone curves in Figs 4 through 10, and the corresponding maximum and minimum levels

of response, are summarized in Table 3 and the percentage change for all the seven buildings

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San Fernando(1971)

Northridge(1994)

Northridge aft. -1

N-S (Longitudinal)R

ocki

ng a

ngle

(rad

)

10-4

10-2

10-3

0.4 0.6 0.80.2

θmax

fmaxfmin

θmin

Zero-crossing method

San Fernando(1971)

Northridge(1994)

Northridgeaft. -1

N-S (Longitudinal)

0.4 0.6 0.80.2

2σf

Gabor transform method

San Fernando(1971)Northridge

(1994)

Northridgeaft. -1

E-W (Transverse)

Instantaneous frequency, fp (Hz)

0.1 0.40.2 0.3

Roc

king

ang

le (r

ad)

10-4

10-2

10-3

θmax

fmin fmax

θmin


San Fernando(1971)

Northridge(1994)

Northridgeaft. -1

E-W (Transverse)


0.40.2 0.50.3

2σf


USGS 0466: Sherman Oaks, 15250 Ventura Blvd.

Figure 4. Instantaneous frequency versus amplitude of motion for station USGS 466.

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USGS 5108: Canoga Park, Santa Susana - ETEC Bldg #462

Northridge(1994)

Aft. 7Aft. 9

Aft. 100

Aft. 129

Aft. 142

Aft. 253

Aft. 254

Aft. 336

Aft. 392

Roc

king

ang

le (r

ad)

10-5

10-6

10-3

10-4

1.0 2.0 2.51.5

E-W Comp.θ

max

fmin fmax

θmin

Zero-crossing method Northridge(1994)

Aft. 7

Aft. 9

Aft. 100

Aft. 129

Aft. 142

Aft. 253

Aft. 254

Aft. 336

Aft. 392

E-W Comp.

2.0 3.02.51.5

2σf


Northridge(1994)

Aft. 9

Aft. 129

Aft. 142

Aft. 151

Aft. 253

Aft. 336

Aft. 392

Roc

king

ang

le (r

ad)

10-5

10-6

10-3

10-4


1.0 2.01.5 2.5

N-S Comp.

θmin

fmin fmax

Zero-crossing methodθ

max

Northridge(1994)

Aft. 9

Aft. 129Aft. 142

Aft. 151

Aft. 253Aft. 336 Aft. 392

N-S Comp.


2.01.5 2.5

2σf



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USGS 5450: Burbank, 3601 West Olive Ave.

Northridge(1994)

Aft. -1

Aft. -5

Aft. -13

Aft. -14

N00E

0.6 0.80.4

Roc

king

ang

le (r

ad)

10-3

10-5

10-4

10-2

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -1

Aft. -5

Aft. -13

Aft. -14

N00E

0.6 0.80.4 1.0

2σf

Gabor transformmethod

Northridge(1994)

Aft. -1

Aft. -5

Aft. -13

Aft. -14

W00N

Roc

king

ang

le (r

ad) 10-3

10-5

10-4

10-2


0.6 0.80.4

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -1

Aft. -5

Aft. -13

Aft. -14

W00N


0.6 0.80.4 1.0

2σf



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USGS 5451: Los Angeles, 6301 Owensmouth Blvd.

Northridge(1994)

Aft. -26

Aft.-115

N00E (Transverse)R

ocki

ng a

ngle

(rad

)

10-4

10-3

10-2

0.2 0.3 0.4

θmax

fmin fmax

θmin


Northridge(1994)

Aft. -26

Aft. -115

N00E (Transverse)

0.50.2 0.3 0.4

2σf


Northridge(1994)

Aft. -26

Aft. -115

W00N (Longitudinal)

Apparent frequency, fp (Hz)

0.3 0.4 0.5

Roc

king

ang

le (r

ad)

10-3

10-4

10-2 θmax

fmin fmax

θmin


Northridge(1994)

Aft. -26

Aft. -115

W00N (Longitudinal)


0.3 0.4 0.5

2σf



16

USGS 5453: Van Nuys, 5805 Sepulveda Blvd.

Northridge(1994)

Aft. -1

Aft. -7

Aft. -29

N00ER

ocki

ng a

ngle

(rad

)

10-4

10-2

10-3

0.80.40.3 0.60.5 0.7

θmax

fminfmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -1

Aft. -7

Aft. -29

N00E

0.80.40.3 0.90.60.5 0.7

2σf

Northridge(1994)

Aft. -7

Aft. -24

Aft. -26

Aft. -29

Aft. -103

Aft. -104

Aft. -115

W00N

Roc

king

ang

le (r

ad)

10-4

10-2

10-3


0.70.5 0.6 0.8 0.9

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -7

Aft. 24

Aft. -26Aft. -29

Aft. -103

Aft. -104

Aft. -115

W00N


0.7 1.00.6 0.8 0.9

2σf




17

USGS 5455: Los Angeles, 16000 Ventura Blvd.

Northridge(1994)

Aft. -1

Aft. -7

Aft. -25Aft. -46

E30S

0.3 0.60.4 0.5

Roc

king

ang

le (r

ad) 10-3

10-4

10-2

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -1

Aft. -7

Aft. -25

Aft. -46

E30S

0.3 0.60.4 0.70.5

2σf


Northridge(1994)

Aft. -1

Aft.-22

Aft. -25

Aft. -46

N30E


0.3 0.60.4 0.50.2

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. -1

Aft. -22

Aft. -25

Aft. -46

N30E


0.3 0.60.4 0.5

2σf


Roc

king

ang

le (r

ad) 10-3

10-4

10-2


18

USGS 5457: Los Angeles, 4836 West 3rd St.

Northridge(1994)

Aft. 1

Aft. 4 Aft. 8

Aft. 9

Aft. 10

Aft. 13

Aft. 19

N00ER

ocki

ng a

ngle

(rad

)

10-4

10-3

10-2

10-5

0.4 0.6 0.8

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. 1

Aft. 4 Aft. 8

Aft. 9

Aft. 10 Aft. 13

Aft. 19

N00E

1.00.4 0.6 0.8

2σf


Northridge(1994)

Aft. 4

Aft. 8

Aft. 9

Aft. 10Aft. 17

Aft. 19

S90W


1.00.5 1.250.75

Roc

king

ang

le (r

ad)

10-4

10-5

10-2

10-3

θmax

fmin fmax

θmin

Zero-crossingmethod

Northridge(1994)

Aft. 4

Aft. 8

Aft. 9Aft. 10

Aft. 17Aft. 19

S90W


1.00.5 1.250.75

2σf



19

Table 3. Maximum and minimum system frequencies and maximum and minimum rocking

angles for seven instrumented buildings.

Station no. Comp.

fmax, fmin (Hz)

∆f / fmax

(%) θmax, θmin

(×10-3 rad) Comp.

fmax, fmin (Hz)

∆f / fmax

(%) θmax, θmin

(×10-3 rad)

5108 E00S 2.130, 1.648 22.64 0.49607, 0.00251 N00E 1.899, 1.525 19.68 1.05640, 0.00395

0466 N00E 0.377, 0.312 17.23 4.74628, 0.12339 W00N 0.295, 0.215 27.23 4.66436, 0.31591

5450 N00E 0.691, 0.614 11.16 3.08807, 0.03879 W00N 0.666, 0.576 13.52 5.16651, 0.03820

5451 N00E 0.329, 0.273 17.16 7.38386, 0.18001 W00N 0.434, 0.373 14.14 9.57349, 0.13386

5453 N00E 0.613, 0.434 29.20 7.87023, 0.06008 W00N 0.744, 0.712 5.69 4.88160, 0.02566

5455 E30S 0.434, 0.408 3.86 5.39350, 0.05552 N30E 0.425, 0.363 14.59 5.36059, 0.09933

5457 N00E 0.675, 0.569 15.76 6.34016, 0.02940 S00W 0.866, 0.704 18.62 3.62546, 0.01286

50

0

10

30

20

40

∆f

f max

(%)

USG

S 51

08

US

GS

046

6

US

GS

545

0

US

GS

545

1 US

GS

545

3

USG

S 54

57

US

GS

545

5

Figure 11. A summary of the changes of the building-soil system frequencies of the seven buildings analyzed in this paper, determined from the observed trends during multiple earthquake excitations (see Figures 4.1 through 4.24). For each building, two values are shown, corresponding to the two horizontal components of motion. The change is expressed as a percentage of the maximum frequency.

is shown in Fig. 11. It is seen that, for these levels of response, the change for most of the

buildings is not more than 20%, but it reaches 30% for two of the buildings.

20

SUMMARY AND CONCLUSIONS

This paper presents results for apparent building frequency for seven buildings in the Los

Angeles area that have recorded the 1994 Northridge earthquake and some of its aftershocks.

Although the number of recorded aftershocks in these buildings was large (up to about 80),

only a small number of records were found to be useable for this analysis, because of the

small signal to noise ratio at long periods which lead to high lower cut-off frequency, higher

or too close to the system frequency.

The system frequency was estimated by two methods—zero crossing and Gabor

analyses. The results by both methods are consistent. The general observed trend of the

variation of the system frequency is decrease during the 1994 Northridge main event, and the

1971 San Fernando earthquake, which caused the largest amplitude response. For all but one

building, the frequency was again larger during the aftershocks, indicating system recovery.

For most buildings, the frequency changed up to 20%, and for two buildings, the change was

about 30%. A permanent reduction of the frequency is consistent with permanent loss of

stiffness, while a “recovery” to the initial higher value is consistent with the interpretation

that the change was mainly due to changes in the soil (rather than in the structure itself), or

changes in the bond between the soil and the foundation. Other possible causes of the

temporary changes are: contribution of the nonstructural elements to the total stiffness in

resisting the seismic forces, and opening of existing cracks in the concrete structures during

larger amplitude response. The degree to which each of these causes contributed to the

temporary changes cannot be determined from the current instrumentation and is beyond the

scope of this analysis. What matters for the design codes, however, is the overall effect,

which can be estimated even from the minimum (roof only) building instrumentation.

ACKNOWLEDGEMENTS

This work was supported by U.S. Geological Survey External Research Program (Grant

No. 03HQGR0013). All views presented are solely those of the authors and do not

necessarily represent the official views of the U.S. Government. The authors are also

grateful to Chris Stephens of the U.S. Geological Survey for kindly supplying film records

for this project from the National Strong Motion Program archives.

21

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